PHP, 46 bytes

No PHP answer on this yet, so here's the easy one:

<?=array_sum(array_map(ord,str_split($argn)));

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$ echo Hello World!|php -F addascii.php
1085

More non-competing answers:

48 bytes

Iterative:

foreach(str_split($argn)as$s)$x+=ord($s);echo$x;

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51 bytes

Iterative 2:

while($i<strlen($argn))$x+=ord($argn[$i++]);echo$x;

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63 bytes

Recursive:

function f($s,$c=0){return$s?f(substr($s,1),ord($s[0])+$c):$c;}

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69 bytes

Using PHP's rarely used _{[citation needed]} array_reduce() function:

<?=array_reduce(str_split($argn),function($c,$i){return$c+ord($i);});

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